Search Previous Year Questions

Given:

324 × 24 = 7776

Formula used:

Dividend ÷ Divisor = Quotient

Calculations:

From the given data:

⇒ 7776 ÷ 24 = 324

We need to find the value of:

⇒ 7.776 ÷ 2.4

Multiplying both the dividend and divisor by 10 to simplify the decimal:

⇒ (7.776 × 10) ÷ (2.4 × 10) = 77.76 ÷ 24

Expressing 77.76 as a fraction:

⇒ 77.76 ÷ 24 = (7776 ÷ 100) ÷ 24

Rearranging the terms:

⇒ (7776 ÷ 24) ÷ 100

Substituting the known value 7776 ÷ 24 = 324:

⇒ 324 ÷ 100 = 3.24

∴ The correct answer is option (1).

Shortcut Trick

Apply the alligation method using the average weights of boys (65) and girls (45) with the class mean (57).

Ratio of Boys to Girls = (57 − 45) : (65 − 57) = 12 : 8

Simplifying the ratio: 12 : 8 = 3 : 2 (Boys : Girls)

The question asks for the ratio of Girls to Boys, which is 2 : 3.

∴ The correct answer is 2:3.

Alternate Method

Given:

Average weight of boys = 65 kg

Average weight of girls = 45 kg

Average weight of the whole class = 57 kg

Formula Used:

Total Weight = (Average Weight × Number of Students)

Let the number of boys be 'B' and girls be 'G'.

⇒ Total weight of class = 57 × (B + G)

⇒ Also, Total weight = 65B + 45G

⇒ 57B + 57G = 65B + 45G

⇒ 57G − 45G = 65B − 57B

⇒ 12G = 8B

⇒ G ÷ B = 8 ÷ 12

⇒ G : B = 2 : 3

∴ The correct answer is 2:3.

Additional Information

Alligation Rule

It is a rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of a specified price: (Quantity of Cheaper) ÷ (Quantity of Dearer) = (Price of Dearer − Mean Price) ÷ (Mean Price − Price of Cheaper).

Weighted Average Formula

The average of a combined group is given by A_m = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂), where n represents the quantity and A represents the average of each group.

Shortcut Trick

Total amount paid using the successive multiplier method:

Net Selling Price = 900 × (90/100) × (95/100)

Net Selling Price = 900 × 0.9 × 0.95

Net Selling Price = 810 × 0.95 = ₹769.50

∴ The correct answer is ₹769.50.

Alternate Method

Given: Marked Price (MP) = ₹900, First Discount = 10%, Additional Discount = 5%.

Step-by-step calculation:

⇒ First discounted price = 900 − (10% of 900)

⇒ First discounted price = 900 − 90 = ₹810

⇒ Final amount paid = 810 − (5% of 810)

⇒ Final amount paid = 810 − 40.50 = ₹769.50

∴ The correct answer is ₹769.50.

Additional Information

Single Equivalent Discount

For two successive discounts of a% and b%, the single equivalent discount is (a + b − ab/100)%.

Marked Price (MP) and Selling Price (SP)

Selling Price is calculated as SP = MP × (100 − Discount%)/100. For successive discounts, multiply each reduction factor.

Successive Discounts with Fraction

If discounts are 10% (1/10) and 5% (1/20), the price becomes 9/10 and 19/20 of the original value respectively.

Shortcut Trick

Initial radius ratio R₁ : R₂ = 2 : 3

New R₁ = 2 × 150% = 3 units

New R₂ = 3 × 120% = 3.6 units

New Radius Ratio = 3 : 3.6 = 5 : 6

Volume Ratio = (Radius Ratio)³ = 5³ : 6³ = 125 : 216

∴ The correct answer is 125 : 216.

Alternate Method

Given:

Ratio of radii R₁ : R₂ = 2 : 3

Increase in R₁ = 50%

Increase in R₂ = 20%

Formula Used:

Volume of a sphere (V) = (4/3) × π × R³

Calculations:

Let the initial radii be R₁ = 2x and R₂ = 3x

⇒ New radius R'₁ = 2x + 50% of 2x = 2x + x = 3x

⇒ New radius R'₂ = 3x + 20% of 3x = 3x + 0.6x = 3.6x

⇒ Ratio of new volumes V'₁ : V'₂ = ((4/3)π(R'₁)³) : ((4/3)π(R'₂)³)

⇒ V'₁ : V'₂ = (R'₁)³ : (R'₂)³

⇒ V'₁ : V'₂ = (3x)³ : (3.6x)³

⇒ V'₁ : V'₂ = 27 : 46.656

⇒ Simplify the ratio: (3 / 3.6)³ = (30 / 36)³

⇒ (5 / 6)³ = 125 : 216

∴ The correct answer is 125 : 216.

Additional Information

Volume of Sphere

The volume of a sphere is given by V = (4/3)πr³, where r is the radius.

Surface Area of Sphere

The total surface area of a sphere is S = 4πr².

Ratio Property

If the ratio of the radii of two spheres is a : b, then the ratio of their surface areas is a² : b² and the ratio of their volumes is a³ : b³.

Shortcut Trick

Total Effective CP = 8000 + 500 = ₹8500

Net multiplier for 20% profit and 25% loss = 1.20 × 0.75 = 0.9

Final Price = 8500 × 0.9 = ₹7650

∴ The correct answer is ₹7650.

Alternate Method

Given:

Purchase price of mobile phone = ₹8000

Amount spent on repairs = ₹500

Profit percentage = 20%

Loss percentage of the buyer = 25%

Formula Used:

Total CP = Cost Price + Overhead Expenses

Selling Price (SP) = CP × (100 + Profit%)/100 or CP × (100 − Loss%)/100

Calculations:

⇒ Total CP for the trader = 8000 + 500 = ₹8500

⇒ Selling price of the trader (SP₁) = 8500 + 20% of 8500

⇒ SP₁ = 8500 + (20/100) × 8500 = 8500 + 1700 = ₹10200

⇒ This SP₁ becomes the Cost Price for the first buyer (CP₂ = ₹10200)

⇒ Final Selling Price (SP₂) = 10200 − 25% of 10200

⇒ SP₂ = 10200 − (25/100) × 10200 = 10200 − 2550 = ₹7650

∴ The correct answer is ₹7650.

Additional Information

Overhead Expenses

Any additional costs like repairs, transportation, or installation must be added to the initial purchase price to determine the actual Cost Price.

Successive Percentage Changes

When a value changes by x% and then y% successively, the net percentage change is [x + y + (xy/100)]%.

Successive Multiplication Factor

Final Value = Initial Value × (1 ± r₁/100) × (1 ± r₂/100) × ... where 'r' is the percentage profit (+) or loss (−).

Shortcut Trick

Check options based on the given conditions.

Condition 1: Denominator − Numerator = 8. For 27/35, 35 − 27 = 8 (Satisfied).

Condition 2: (Numerator + 3) / (Denominator + 5) = 3/4.

⇒ (27 + 3) / (35 + 5) = 30 / 40 = 3/4 (Satisfied).

∴ The correct answer is 27/35.

Alternate Method

Given:

Numerator = Denominator − 8

New Numerator = Numerator + 3

New Denominator = Denominator + 5

New Fraction = 3/4

Calculations:

Let the denominator of the original fraction be x.

⇒ Numerator = x − 8

⇒ Original Fraction = (x − 8) / x

According to the question:

⇒ ((x − 8) + 3) / (x + 5) = 3 / 4

⇒ (x − 5) / (x + 5) = 3 / 4

By cross-multiplication:

⇒ 4 × (x − 5) = 3 × (x + 5)

⇒ 4x − 20 = 3x + 15

⇒ 4x − 3x = 15 + 20

⇒ x = 35

Now, find the numerator:

⇒ Numerator = 35 − 8 = 27

⇒ Original Fraction = 27/35

∴ The correct answer is 27/35.

Additional Information

Proper and Improper Fractions

A Proper Fraction is one where the numerator is less than the denominator (e.g., 27/35). An Improper Fraction has a numerator greater than or equal to the denominator.

Equivalent Fractions

Fractions that represent the same value even though they look different, obtained by multiplying or dividing the numerator and denominator by the same non-zero number.

Cross-Multiplication Rule

In an equation like a/b = c/d, the property a × d = b × c is used to solve for unknown variables in linear equations involving fractions.

Shortcut Trick

Distance between two objects on the same side = Height × (cot θ_smaller − cot θ_larger)

Distance = 75 × (cot 30° − cot 45°)

Distance = 75 × (√3 − 1) = 75 × (1.732 − 1)

Distance = 75 × 0.732 = 54.9 m

∴ The correct answer is 54.9 m.

Alternate Method

Given:

Height of the lighthouse (AB) = 75 m

Angles of depression of two ships = 30° and 45°

Since the ships are on the same side, the angles of elevation from the ships to the top are also 30° and 45°.

Calculations:

⇒ In right ΔABC (closer ship at C):

⇒ tan 45° = AB ÷ BC

⇒ 1 = 75 ÷ BC ⇒ BC = 75 m

⇒ In right ΔABD (farther ship at D):

⇒ tan 30° = AB ÷ BD

⇒ 1 ÷ √3 = 75 ÷ BD ⇒ BD = 75√3 m

⇒ Distance between ships (CD) = BD − BC

⇒ CD = 75√3 − 75 = 75(√3 − 1)

⇒ CD = 75 × (1.732 − 1) = 75 × 0.732

⇒ CD = 54.9 m

∴ The correct answer is 54.9 m.

Additional Information

Angle of Depression

The angle formed by the line of sight with the horizontal line for an object below the horizontal level.

Trigonometric Ratios

For height and distance problems, tan θ = Perpendicular ÷ Base and cot θ = Base ÷ Perpendicular are most commonly used.

General Formula for Same Side

Distance d = h × |cot α − cot β|, where h is height and α, β are angles of elevation/depression.

Given:

((13/21) / (13/10)) ÷ ((5/9 × 18/15) + 5/7) + 10/6 ÷ (29/15 of 15/6)

Formula used:

The VBODMAS rule (Vinculum, Bracket, Of, Division, Multiplication, Addition, Subtraction) is used to simplify the expression.

Calculations:

((13/21) × (10/13)) ÷ ((1/3 × 2/1) + 5/7) + 10/6 ÷ (29/15 × 15/6)

⇒ (10/21) ÷ (2/3 + 5/7) + 10/6 ÷ 29/6

⇒ (10/21) ÷ ((14 + 15) / 21) + (10/6 × 6/29)

⇒ (10/21) ÷ (29/21) + 10/29

⇒ (10/21 × 21/29) + 10/29

⇒ 10/29 + 10/29

⇒ 20/29

∴ The correct answer is option (2).

Shortcut Trick

Using the Net Deviation method where the Average is 15.

Deviations of given numbers from 15: (10 − 15) = −5, (20 − 15) = +5, (25 − 15) = +10.

Sum of known deviations = −5 + 5 + 10 = +10.

To keep the average 15, the total deviation must be 0; thus, the fourth number deviation = −10.

Fourth number = 15 − 10 = 5.

∴ The correct answer is 5.

Alternate Method

Given:

Average of 4 numbers = 15

Three numbers are 10, 20, and 25.

Formula Used:

Average = (Sum of all observations) ÷ (Total number of observations)

Calculations:

⇒ Let the fourth number be x.

⇒ Average = (10 + 20 + 25 + x) ÷ 4

⇒ 15 = (55 + x) ÷ 4

⇒ 15 × 4 = 55 + x

⇒ 60 = 55 + x

⇒ x = 60 − 55

⇒ x = 5

∴ The correct answer is 5.

Additional Information

Sum of Observations

The total sum is always equal to Average × Number of items (Sum = A × n).

Net Deviation Rule

The algebraic sum of deviations of all observations from their arithmetic mean is always zero.

Change in Average

If a constant k is added to or subtracted from every observation, the new average will be (Old Average ± k).

Shortcut Trick

Let the Marked Price (MP) be 100 units.

Cost Price (CP) for Ravi = 100 − 25% of 100 = 75 units.

Ravi sells it at 10% profit ⇒ Selling Price (SP) = 75 × 110 ÷ 100 = 82.5 units.

Given SP = ₹1,320 ⇒ 82.5 units = 1320.

1 unit = 1320 ÷ 82.5 = 16.

Marked Price = 100 units = 100 × 16 = ₹1,600.

∴ The correct answer is Rs. 1600.

Alternate Method

Given: Discount = 25%, Profit = 10%, Selling Price (SP) = ₹1,320.

Formulas: CP = SP × 100 ÷ (100 + Profit%); MP = CP × 100 ÷ (100 − Discount%)

⇒ First, find the Cost Price (CP) for Ravi: CP = 1320 × 100 ÷ 110 = ₹1,200.

⇒ This CP is the price after a 25% discount on the Marked Price (MP).

⇒ Let MP be x. Then, 75% of x = 1200.

⇒ x = (1200 × 100) ÷ 75 = 16 × 100 = ₹1,600.

∴ The correct answer is Rs. 1600.

Additional Information

Marked Price & Discount

The Marked Price (MP) is the price printed on an item. Discount is always calculated on the MP to arrive at the Selling Price (SP).

Relation between CP and MP

CP ÷ MP = (100 − Discount%) ÷ (100 + Profit%). This direct ratio helps find MP when CP and profit/discount percentages are known.